Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey

Na Min; Mingxin Wang

doi:10.3934/dcdsb.2018073

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2018, Volume 23, Issue 4: 1721-1737. Doi: 10.3934/dcdsb.2018073

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Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey

Na Min and
Mingxin Wang^,

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Mingxin Wang
* Corresponding author: Mingxin Wang

Received: July 31, 2017

Early access: January 2018

Published: April 2018

This work was supported by NSFC Grant 11371113.

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Abstract

In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

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Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 35B32, 35B35, 35B40, 35K57.

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